A Random Trotter Product Formula

Let be a pure jump process with state space S and let <!-- MATH ${\xi _0},{\xi _1},{\xi _2}, \cdots$ --> be the succession of states visited by <!-- MATH $X(t),{\Delta _0}{\Delta _1} \cdots$ --> the sojourn times in each state, the number of transitions before t and <!-- MATH ${\Delta _t} = t - \sum\nolimits_{k = 0}^{N(t) - 1} {{\Delta _k}}$ --> . For each let be an operator semigroup on a Banach space L. Define <!-- MATH ${T_\lambda }(t,w) = {T_{{\xi _0}}}((1/\lambda ){\Delta _0}){T_{{\xi _1}}}((1/\lambda ){\Delta _1}) \cdots T_{\xi _{N(\lambda t)}}((1/\lambda ){\Delta _{\lambda t}})$ --> . Conditions are given under which <!-- MATH ${T_\lambda }(t,w)$ --> will converge almost surely (or in probability) to a semigroup of operators as <!-- MATH $\lambda \to \infty$ --> . With <!-- MATH $S = \{ 1,2\}$ --> and <!-- MATH \begin{displaymath} \begin{array}{*{20}{c}} \hfill {X(t) = 1,\quad } \\\hfill { = 2,\quad } \\\end{array} \begin{array}{*{20}{c}} {2n \leqq t < 2n + 1,} \hfill \\{2n + 1 \leqq t < 2n + 2,} \hfill \\\end{array} \end{displaymath} -->